Energy levels A,B,C of a certain atoms corresponding to increasing values of energy level i.e., `E_(A) lt E_(B) lt E_(C)`. If `lambda_(1), lambda_(2)` and `lambda_(3)` are the wavelengths of radiations corresponding to the transitions C to B,B to A and C to A respectively which of the following statement is correct?

Options

A. `lambda_(3) = lambda_(1) +lambda_(2)`
B. `lambda_(3) = (lambda_(1)lambda_(2))/(lambda_(1)+lambda_(2))`
C. `lambda_(1)+lambda_(2)+ lambda_(3) = 0`
D. `lambda_(3)^(2) =lambda_(1)^(2)+lambda_(2)^(2)`

Explanation

To solve the problem, we need to analyze the transitions between the energy levels of the atom and relate them to the wavelengths of the emitted radiation.

### Step-by-Step Solution:

1. **Identify Energy Levels**:
- We have three energy levels: A, B, and C, where \( E_A < E_B < E_C \). This means that energy level A has the lowest energy, followed by B, and C has the highest energy.

2. **Define the Wavelengths**:
- Let \( \lambda_1 \) be the wavelength of the radiation corresponding to the transition from C to B.
- Let \( \lambda_2 \) be the wavelength of the radiation corresponding to the transition from B to A.
- Let \( \lambda_3 \) be the wavelength of the radiation corresponding to the transition from C to A.

3. **Apply Energy-Wavelength Relationship**:
- The energy of a photon is related to its wavelength by the equation:
\[ E = \frac{hc}{\lambda} \]
- Where \( h \) is Planck's constant and \( c \) is the speed of light.

4. **Write Energy Equations**:
- For the transition C to B:
\[ E_{CB} = E_C - E_B = \frac{hc}{\lambda_1} \]
- For the transition B to A:
\[ E_{BA} = E_B - E_A = \frac{hc}{\lambda_2} \]
- For the transition C to A:
\[ E_{CA} = E_C - E_A = \frac{hc}{\lambda_3} \]

5. **Relate the Energies**:
- From the above equations, we can express the energy of the transition C to A in terms of the other two transitions:
\[ E_{CA} = E_{CB} + E_{BA} \]
- Substituting the energy expressions:
\[ \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \]

6. **Simplify the Equation**:
- Cancel \( hc \) from both sides:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]

7. **Final Relationship**:
- Rearranging gives:
\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2} \]
- This indicates that the wavelength \( \lambda_3 \) is related to \( \lambda_1 \) and \( \lambda_2 \) through the formula derived.

### Conclusion:
The correct statement is that the relationship between the wavelengths is given by:
\[ \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]